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The set of all points where the function f(x)=x+|x| is differentiable, is

(a) (0,∞)                                    (b) (-∞,0)

(c) (-∞,0)∪(0,∞)                        (d) (-∞,∞)

This question is similar to Question-15 NCERT-Exemplar-MCQs


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๐‘“(๐‘ฅ) = ๐‘ฅ+|๐‘ฅ| = {โ–ˆ(๐‘ฅ+๐‘ฅ, ๐‘ฅโ‰ฅ0@๐‘ฅโˆ’๐‘ฅ, ๐‘ฅ <0)โ”ค = {โ–ˆ(2๐‘ฅ, ๐‘ฅโ‰ฅ0@0, ๐‘ฅ<0)โ”ค Now, f(x) is a differentiable at x = 0 if LHD = RHD (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™) โˆ’ ๐’‡(๐’™ โˆ’ ๐’‰))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0) โˆ’ ๐‘“(0 โˆ’ โ„Ž))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐ŸŽ) โˆ’ ๐’‡(โˆ’๐’‰))/๐’‰ =(๐‘™๐‘–๐‘š)โ”ฌ(โ„Žโ†’0) (2(0) โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = ๐ŸŽ (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™ + ๐’‰) โˆ’ ๐’‡(๐’™ ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0 + โ„Ž) โˆ’ ๐‘“(0))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’‰) โˆ’ ๐’‡(๐ŸŽ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2โ„Ž โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 2โ„Ž/โ„Ž = ๐Ÿ Hence, we can say that f(x) is differentiable on R โˆ’ {๐ŸŽ} or (โˆ’โˆž,๐ŸŽ)โˆช(๐ŸŽ,โˆž) So, the correct answer is (c)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.